Banach-Lie groupoids associated to \(W^*\)-algebras (Q344359)

The purpose of this paper is to highlight certain natural connections between Banach-Poisson geometry, groupoid theory and \(W^*\)-algebras. The motivation is given by the existence of the canonically defined Banach-Lie-Poisson structure on the predual of a \(W^*\)-algebra illustrated in [\textit{A. Odzijewicz} and \textit{T. S. Ratiu}, Comm. Math. Phys. 243, 1--54 (2003; Zbl 1044.53057)], and by the significance of this structure for the theory of infinite-dimensional Hamiltonian systems pointed out in [\textit{A. Odzijewicz} and \textit{T. S. Ratiu}, J. Funct. Anal. 255, No. 5, 1225--1272 (2008; Zbl 1157.53044)]. The authors associate to a \(W^*\)-algebra \(M\) various groupoids (the groupoid consisting of the partially invertible elements of \(M\), the groupoid of the partial isometries in \(M\), as well as action groupoids related to it) and describe the relationship between these groupoids and the Banach-Poisson geometry.